Bregman Voronoi Diagrams: Properties, Algorithms and Applications

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures

Subdivision surface fitting to a dense mesh using ridges and umbilics

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach

Path Planning for Mobile Robot Navigation using Voronoi Diagram and Fast Marching

For navigation in complex environments, a robot need s to reach a compromise between the need for having efficient and optimized trajectories and t he need for reacting to unexpected events. This paper presents a new sensor-based Path Planner w hich results in a fast local or global motion planning able to incorporate the new obstacle information. In the first step the safest areas in the environment are extracted by means of a Vorono i Diagram. In the second step the Fast Marching Method is applied to the Voronoi extracted a reas in order to obtain the path. The method combines map-based and sensor-based planning o perations to provide a reliable motion plan, while it operates at the sensor frequency. The m ain characteristics are speed and reliability, since the map dimensions are reduced to an almost uni dimensional map and this map represents the safest areas in the environment for moving the robot. In addition, the Voronoi Diagram can be calculated in open areas, and with all kind of shaped obstacles, which allows to apply the proposed planning method in complex environments wher e other methods of planning based on Voronoi do not work.This work has been supported by the CAM Project S2009/DPI-1559/ROBOCITY2030 I

Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension

In binary images, the distance transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for $d$-dimensional images. We also present a $d$-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

Efficient Distance Transformation for Path-based Metrics

In many applications, separable algorithms have demonstrated their efficiency to perform high performance volumetric processing of shape, such as distance transformation or medial axis extraction. In the literature, several authors have discussed about conditions on the metric to be considered in a separable approach. In this article, we present generic separable algorithms to efficiently compute Voronoi maps and distance transformations for a large class of metrics. Focusing on path-based norms (chamfer masks, neighborhood sequences...), we propose efficient algorithms to compute such volumetric transformation in dimension $n$. We describe a new $O(n\cdot N^n\cdot\log{N}\cdot(n+\log f))$ algorithm for shapes in a $N^n$ domain for chamfer norms with a rational ball of $f$ facets (compared to $O(f^{\lfloor\frac{n}{2}\rfloor}\cdot N^n)$ with previous approaches). Last we further investigate an even more elaborate algorithm with the same worst-case complexity, but reaching a complexity of $O(n\cdot N^n\cdot\log{f}\cdot(n+\log f))$ experimentally, under assumption of regularity distribution of the mask vectors

Parallel Voronoi Computation for Physics-Based Simulations

International audienceVoronoi diagrams are fundamental data structures in computational geometry, with applications in such areas as physics-based simulations. For non-Euclidean distances, the Voronoi diagram must be performed over a grid-graph, where the edges encode the required distance information. Th e major bottleneck in this case is a shortest path algorithm that must be computed multiple times during the simulation. We present a GPU algorithm for solving the shortest path problem from multiple sources using a generalized distance function. Our algorithm was designed to leverage the grid-based nature of the underlying graph that represents the deformable objects. Experimental results report speed-ups up to 65× over a current reference sequential method

Mobile robot path planning using Voronoi diagram and fast marching

For navigation in complicated environments, a robot must reach a compromise between efficient trajectories and ability to react to unexpected environmental events. This paper presents a new sensorbased path planner, which gives a fast local or global motion plan capable to incorporate new obstacles data. Within the first step, the safest areas in the environment are extracted by means of a Voronoi Diagram. Within the second step, the fast marching method is applied to the Voronoi extracted areas so as to get the trail. This strategy combines map-based and sensor-based designing operations to supply a reliable motion plan, whereas it operates at the frequency of the sensor. The most interesting characteristics are high speed and reliability, as the map dimensions are reduced to a virtually one-dimensional map and this map represents the safest areas within the environment. Additionally, the Voronoi Diagram is calculated in open areas with all reasonably shaped obstacles. This fact permits to use the planned trajectory methodology in complex environments wherever different Voronoi-based strategies will not work.Publicad